Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6359112 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×D42 |
| commutator, Frattini, cosocle | C42 | C25 |
| prime radical | #61329 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
1569 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,6,9)(3,7,5,13)(4,11,12,15)(8,14,16,10)(18,24)(19,26)(21,22)(25,32)(27,28)(29,30);(1,3,6,5)(2,7,9,13)(4,10,12,14)(8,15,16,11)(17,18)(19,23)(20,28)(21,26)(22,24)(25,30)(27,29)(31,32);(1,4,6,12)(2,8,9,16)(3,10,5,14)(7,15,13,11)(17,19,23,18)(20,27,31,25)(21,26,22,24)(28,30,32,29);(17,20)(18,25)(19,27)(21,29)(22,30)(23,31)(24,32)(26,28);(17,21,23,22)(18,26,19,24)(20,29,31,30)(25,28,27,32)Polycyclic presentation:
2^9|a^2=hi,b^2=hi,c^2=i,e^2=h,f^2=h,g^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fh,g^a=gi,c^b=ch,d^b=df,f^b=fh,e^c=eh,g^c=gi,f^d=fh86 conjugacy classes — view the full character table.
| maximal subgroup of | #72283373 |
| maximal normal subgroup of | #72283373 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6266362);